Difficult integration

**mikegar813** · April 13th 2009, 12:48 PM

Evaluate $2\pi \int_{\frac{\pi}{8}}^{\frac{3\pi}{8}} \tan x + \sqrt{1 + \sec ^4 x} dx$

i know the final anwser i need to know how to do it. its not for homework or a test i just want to know how to do it

**Jhevon** · April 13th 2009, 01:15 PM

lets concentrate on $\int \tan x \sqrt{1 + \sec^4 x}~dx$

Note that this is the same as $\int \frac {\sec^4x \tan x \sqrt{1 + \sec^4 x}}{\sec^4 x}~dx$ ........i multiplied by $\frac {\sec^4 x}{\sec^4 x}$

Now, let $u^2 = 1 + \sec^4 x$

$\Rightarrow 2u~du = 4 \sec^4 x \tan x~dx$

and $\sec^4 x = u^2 - 1$

So our integral becomes:

$\frac 12 \int \frac {u^2}{u^2 - 1}~du$

which is relatively easy to deal with

**Soroban** · April 13th 2009, 02:40 PM

Wow! . . . great soluton, Jhevon!

**Krizalid** · April 13th 2009, 03:37 PM

indeed haha! actually, this problem came from http://www.mathhelpforum.com/math-he...plication.html

and look at my long solution, i thought many things at the same time!

**Danny** · April 13th 2009, 03:37 PM

Hey Jhevon - Ever been call a Mathimagican

**Jhevon** · April 13th 2009, 05:44 PM

Originally Posted by Krizalid

indeed haha! actually, this problem came from http://www.mathhelpforum.com/math-he...plication.html

and look at my long solution, i thought many things at the same time!

well, look at that. i never remembered seeing this integral before. how did you?!

it was probably a mistake that i stumbled on it

i just wanted to get rid of the square root. so i made what's underneath it $u^2$ . then i realized that i would need a $sec^4 x$ that wasn't there, so i put it there. wonder how it came to me

Originally Posted by danny arrigo

Hey Jhevon - Ever been call a Mathimagican

haha, no, and for good reason

**matheagle** · April 13th 2009, 06:01 PM

Originally Posted by danny arrigo

Hey Jhevon - Ever been call a Mathimagican

I had a student call me that quite often last year.
I'm not sure what he meant by it.

**Danny** · April 14th 2009, 04:44 AM

Originally Posted by matheagle

I had a student call me that quite often last year.
I'm not sure what he meant by it.

Ya me to, especially with nonlinear differential equations. They keep saying that and "Danny magic."

**matheagle** · April 14th 2009, 07:23 AM

I guess this is magic to them

**mr fantastic** · April 14th 2009, 07:34 AM

Originally Posted by matheagle

I guess this is magic to them

To paraphrase Arthur C. Clarke: Any sufficiently advanced mathematics is indistinguishable from magic.

And to paraphrase a Moderator: This thread is gettting way off-topic. The question has been answered so ....... thread closed.

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